I don't think that it would pose a huge threat over contamination.
It has not been boiled (sterilized) yet. And it's only one day of sitting there.
I don't think there should be any problems with doing this.

I think there's a strong chance that the wort, which has not been sanitized itself, might sour. This is a little too close to how Berliner Weiße is made for me to be comfortable. If I was going to do this, I'd at least bring the wort just to boiling before leaving it overnight.

Well, Berliner Weiße ferments longer than overnight, but that's how a local brewpub makes their excellent Berliner Weiße. I'd be worried that you would still get some sour flavors from lactic acid bacteria working overnight. It might work for a Dry Stout if you wanted that sour background flavor that some people try to achieve.

To make a long story short, a few times I've had unexpected emergencies, etc., arise during a brew day, and have had to postpone boil to the next day.

First, no matter how much you sanitize your kettle or other container you are draining your runnings into, your runnings are not sanitized. The mash tun and lautering equipment (false bottom, manifold, mesh hose) are not sanitized, and the grain that you started with is not sanitized, so there is bound to be bacteria. Although mash-out at about 170F will kill a lot of bacteria, it will not kill all of it.

My first time, I just let the sweet wort sit overnight and it had definitely soured a bit; I boiled it anyway, and the beer was drinkable, but with a noticeable sourness in the background, sort of like a lambic.

My next time, I had enough time to bring it to a boil for a few minutes -- but wouldn't have time for chilling, aeration, etc,, so I then let it sit covered. The next day it was cooled off almost completely but I didn't notice any significant souring; but there were large clumps of what I assume were proteins floating around in it, but the beer was 'okay'. Not as good as I want when conditioned, but drinkable.

The last time I was interrupted, I put the kettle on the kitchen stove and after a brief boil I set the burner on 'low' and just let it keep the kettle and wort hot all night. When I got up in the morning, the wort was fine and I turned up the heat while I set up my propane burner, and so I was able to reach a boil very quickly. I would therefore recommend that last method.

I suppose you could crash cool it and store in the refrigerator, but that takes time (cooling) and sanitizing your chiller, and room in your refrigerator, and will take that much longer to reach a boil the next day.

EDIT: Just an addition note to shaggyt: I think very low temps will probably allow you to keep your wort for several days, and if you are really approaching 32F, you can probably store it for even weeks. If you find yourself in circumstances where you need to interrupt your brew for more than a day, then I would opt for storing it cold rather than trying to keep it hot for several days. The safest way to store cold, in my opinion, is to bring it to a boil first to kill as much bacteria as possible, crash chill, seal very well (I would even consider duct tape or something like that to prevent refrigerator flavors from getting into the wort), and store as cold as possible.